| L(s) = 1 | + 1.93·2-s + 1.74·4-s − 0.508·5-s + 3.68·7-s − 0.491·8-s − 0.983·10-s + 0.318·11-s + 4.18·13-s + 7.12·14-s − 4.44·16-s − 3.17·17-s − 5.87·19-s − 0.887·20-s + 0.616·22-s − 2.50·23-s − 4.74·25-s + 8.10·26-s + 6.42·28-s − 29-s + 2.50·31-s − 7.61·32-s − 6.14·34-s − 1.87·35-s + 7.87·37-s − 11.3·38-s + 0.249·40-s − 8.72·41-s + ⋯ |
| L(s) = 1 | + 1.36·2-s + 0.872·4-s − 0.227·5-s + 1.39·7-s − 0.173·8-s − 0.311·10-s + 0.0960·11-s + 1.16·13-s + 1.90·14-s − 1.11·16-s − 0.769·17-s − 1.34·19-s − 0.198·20-s + 0.131·22-s − 0.522·23-s − 0.948·25-s + 1.59·26-s + 1.21·28-s − 0.185·29-s + 0.450·31-s − 1.34·32-s − 1.05·34-s − 0.316·35-s + 1.29·37-s − 1.84·38-s + 0.0395·40-s − 1.36·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.466756777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.466756777\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.93T + 2T^{2} \) |
| 5 | \( 1 + 0.508T + 5T^{2} \) |
| 7 | \( 1 - 3.68T + 7T^{2} \) |
| 11 | \( 1 - 0.318T + 11T^{2} \) |
| 13 | \( 1 - 4.18T + 13T^{2} \) |
| 17 | \( 1 + 3.17T + 17T^{2} \) |
| 19 | \( 1 + 5.87T + 19T^{2} \) |
| 23 | \( 1 + 2.50T + 23T^{2} \) |
| 31 | \( 1 - 2.50T + 31T^{2} \) |
| 37 | \( 1 - 7.87T + 37T^{2} \) |
| 41 | \( 1 + 8.72T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 3.87T + 61T^{2} \) |
| 67 | \( 1 - 7.04T + 67T^{2} \) |
| 71 | \( 1 + 6.24T + 71T^{2} \) |
| 73 | \( 1 + 7.87T + 73T^{2} \) |
| 79 | \( 1 + 4.85T + 79T^{2} \) |
| 83 | \( 1 - 8.37T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.89379906390817673898495680908, −11.45717177468320183508910298287, −10.50317658005145421373983048345, −8.827804660270956228612202293003, −8.110151513408090971118370390688, −6.64955494139607739085014912079, −5.69070465123847630004924937603, −4.54613745325072059167389947720, −3.85551656049954225721533006205, −2.09940579716822489259011887500,
2.09940579716822489259011887500, 3.85551656049954225721533006205, 4.54613745325072059167389947720, 5.69070465123847630004924937603, 6.64955494139607739085014912079, 8.110151513408090971118370390688, 8.827804660270956228612202293003, 10.50317658005145421373983048345, 11.45717177468320183508910298287, 11.89379906390817673898495680908
